You quickly locate a diagram of the tachyon manifold (your puzzle input). A tachyon beam enters the manifold at the location marked S; tachyon beams always move downward. Tachyon beams pass freely through empty space (.). However, if a tachyon beam encounters a splitter (^), the beam is stopped; instead, a new tachyon beam continues from the immediate left and from the immediate right of the splitter.

For example:

.......S.......
...............
.......^.......
...............
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............

In this example, the incoming tachyon beam (|) extends downward from S until it reaches the first splitter:

.......S.......
.......|.......
.......^.......
...............
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............

At that point, the original beam stops, and two new beams are emitted from the splitter:

.......S.......
.......|.......
......|^|......
...............
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............

Those beams continue downward until they reach more splitters:

.......S.......
.......|.......
......|^|......
......|.|......
......^.^......
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............

At this point, the two splitters create a total of only three tachyon beams, since they are both dumping tachyons into the same place between them:

.......S.......
.......|.......
......|^|......
......|.|......
.....|^|^|.....
...............
.....^.^.^.....
...............
....^.^...^....
...............
...^.^...^.^...
...............
..^...^.....^..
...............
.^.^.^.^.^...^.
...............

This process continues until all of the tachyon beams reach a splitter or exit the manifold:

.......S.......
.......|.......
......|^|......
......|.|......
.....|^|^|.....
.....|.|.|.....
....|^|^|^|....
....|.|.|.|....
...|^|^|||^|...
...|.|.|||.|...
..|^|^|||^|^|..
..|.|.|||.|.|..
.|^|||^||.||^|.
.|.|||.||.||.|.
|^|^|^|^|^|||^|
|.|.|.|.|.|||.|

To repair the teleporter, you first need to understand the beam-splitting properties of the tachyon manifold. In this example, a tachyon beam is split a total of *21* times.

Analyze your manifold diagram. How many times will the beam be split?

Solution

Simple simulation, keeping track of active beams using an unordered_set to avoid counting duplicate beams. Iterate over each row until the bottom.

Part 2